Elastic least-squares reverse time migration

Yuting Duan1, Antoine Guitton1, and Paul Sava1

ABSTRACT

Least-squares migration can produce images with im-proved resolution and reduced migration artifacts, comparedwith conventional imaging. We have developed a method forelastic least-squares reverse time migration (LSRTM) basedon a new perturbation imaging condition that yields scalarimages of squared P- and S-velocity perturbations. Theseperturbation images do not suffer from polarity reversals thatare common for more conventional elastic imaging methods.We use 2D synthetic and field-data examples to demonstratethe proposed LSRTM algorithm using the perturbation im-aging condition. Our results show that elastic LSRTM im-proves the energy focusing and illumination of the elasticimages and it attenuates artifacts resulting, for instance, fromsparseness in the wavefield sampling and crosstalk of theP- and S-modes. Compared with RTM images, the LSRTMimages provide more accurate relative amplitude informa-tion that is useful for reservoir characterization.

INTRODUCTION

Seismic migration is a technique for obtaining structural imagesof the subsurface from recorded seismic data. Starting from a lin-earized forward operator that is based on assumptions about thewave equation and model parameters, seismic migration can be for-mulated as the adjoint operator that maps seismic data to a subsur-face image (Claerbout, 1992). Migrated images not only can showgeologic structures but they also can provide information aboutmaterial properties, such as reflectivity, which is important for res-ervoir characterization. In practice, however, migration imagesoften contain various undesirable artifacts, for example, the artifactscaused by limited bandwidth and acquisition coverage. Also, migra-tion images are usually computed under assumptions about wavepropagation in the subsurface, e.g., that the earth is isotropic and

acoustic; these assumptions do not fully represent the real elasticearth. Taken together, these limitations and assumptions result inartifacts.Advances in seismic acquisition and ongoing improvements in

computational capability make imaging using elastic waves increas-ingly feasible (Sun and McMechan, 1986; Hokstad et al., 1998; Sunet al., 2006; Denli and Huang, 2008; Yan and Sava, 2008; Artmanet al., 2009; Wu and Yan, 2010; Du et al., 2012b; Duan and Sava,2015; Rocha et al., 2016). Compared with acoustic images, elasticimages can provide more information about the subsurface, e.g.,fracture distributions and elastic properties. However, elastic migra-tion suffers from issues that negatively affect the quality of the im-ages. Because it is, in general, difficult to separate all arrivals bywave mode in recorded data, nonphysical modes lead to artifacts(i.e., crosstalk) in the image (Duan and Sava, 2014).Least-squares migration (LSM) is an improved imaging algo-

rithm that reduces these migration artifacts and also improvesthe resolution of migration images. LSM is a linearized waveforminversion that seeks to find the image that best predicts, in a least-squares sense, the recorded seismic data (Schuster et al., 1993;Nemeth et al., 1999; Kuehl and Sacchi, 2002; Kaplan et al.,2010). Schuster et al. (1993) propose LSM for crosswell data,whereas Nemeth et al. (1999) apply this technique to surface data.Their studies show that LSM can significantly improve the spatialresolution of the images and can also reduce migration artifacts aris-ing from limited aperture, coarse sampling, and acquisition gaps.LSM can be implemented using a Kirchhoff engine (Nemeth

et al., 1999; Dai et al., 2011), one-way wave propagator (Kuehland Sacchi, 2002; Kaplan et al., 2010; Huang and Schuster,2012), or two-way wave propagator, i.e., least-squares reverse timemigration (LSRTM) (Dong et al., 2012; Dai and Schuster, 2013;Hou and Symes, 2015; Wong et al., 2015). For elastic LSM, Stantonand Sacchi (2015) propose elastic LSM using a one-way waveequation, and they compute PP and PS images from 2C elastic datain isotropic media. Although computationally expensive, RTM isadvantageous for velocity models with strong lateral variationsor with complicated geologic structures that result in wavefieldmultipathing.

Manuscript received by the Editor 1 November 2016; revised manuscript received 5 February 2017; published online 30 May 2017.1Center for Wave Phenomena, Colorado School of Mines, Golden, Colorado, USA. E-mail: yduan@mines.edu; aguitton@gmail.com; psava@mines.edu.© 2017 Society of Exploration Geophysicists. All rights reserved.

S315

GEOPHYSICS, VOL. 82, NO. 4 (JULY-AUGUST 2017); P. S315–S325, 19 FIGS.10.1190/GEO2016-0564.1

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

A key component for elastic LSRTM is the imaging condition,and many different types of imaging conditions have been proposedfor elastic media. For example, Yan and Sava (2008) propose a dis-placement imaging condition that crosscorrelates each componentof the source and receiver displacement wavefields. They also pro-pose a potential imaging condition that crosscorrelates P- and S-wave modes derived from the source and receiver wavefields.One issue with this potential imaging condition is that the imagecomponents for converted waves change polarity at normal inci-dence. Therefore, they adopt an additional polarity correction inthe angle domain, which is computationally expensive. Duan andSava (2015) propose a scalar imaging condition for converted wavesthat produces scalar images without polarity reversal; however, thisimaging condition requires knowledge of the geologic dip. Du et al.(2012a) reverse the polarity in source and receiver wavefields basedon the directions of the incident and reflected waves. These directionstypically are computed using Poynting vectors, which may be inac-curate in complicated models characterized by multipathing (Dickensand Winbow, 2011; Patrikeeva and Sava, 2013).In this paper, we propose an elastic LSRTM method based on a

new perturbation imaging condition, which is derived for squared Pand S velocities. Images computed using this new imaging condi-tion can be related to physical subsurface properties, and they do notsuffer from polarity changes; they can be stacked over experimentswithout an additional polarity correction, thus reducing the compu-tational cost of the algorithm. Using 2D synthetic and field-dataexamples, we demonstrate that we are able to obtain elastic LSRTMimages with higher resolution and reduced migration artifacts.

THEORY

LSM aims to find the image that best predicts, in a least-squaressense, the recorded seismic data. For elastic migration, we considera vector image m, which contains P- and S-wave lithologic infor-mation. We treat migration as an adjoint operator FT that transformsdata d to imagem, and thus the forward process can be expressed as

Fm ¼ d; (1)

where F is the demigration operator. Data dðe; x; tÞ are a vector, as afunction of the experiment index e, spatial location x, and time t.For LSM, one typically updates the model iteratively by minimiz-

ing the objective function:

J ¼Xe

1

2kWðFm − drÞk2; (2)

which evaluates the misfit between the observed data dr and pre-dicted data ðFmÞ for each experiment e. The operator Wðe; x; tÞdenotes a data weighting operator, which can be applied for variouspurposes. For example, Trad et al. (2015) use a matrix W to elimi-nate the impact of high-amplitude noise or missing traces on inver-sion; Wong et al. (2015) use W to weigh the salt reflection energydown. In this paper, we use the data weighting term to equalize theamplitudes of all arrivals in the recorded data, and thus to obtainbalanced updates for all reflectors.We derive perturbation models using the Born approximation

(Hudson and Heritage, 1981; Jaramillo and Bleistein, 1999; Ribo-detti et al., 2011). We consider the homogeneous elastic isotropicwave equation:

us − α∇ð∇ · usÞ þ β∇ × ð∇ × usÞ ¼ ds: (3)

The background elastic models are assumed to be slowly vary-ing. Vector usðe; x; tÞ ¼ ½ ux uy uz �T is the source-displacementwavefield, which is a function of experiment e, space x, and time t.The vector dsðe; x; tÞ is the source function, which we assume to beknown. The parameters αðxÞ ¼ ðλþ 2μÞ∕ρ and βðxÞ ¼ ðμ∕ρÞ arethe squared P- and S-wave velocities, respectively; λ and μ arethe Lamé parameters, and ρ is the density.The perturbation m ¼ ½ δα δβ �T added to the background

model gives the perturbed model ½ αþ δα β þ δβ �T . Under theBorn approximation, the total wavefield us þ δus is computed us-ing the same source term ds:

ðus þ δusÞ − ðαþ δαÞ∇½∇ · ðus þ δusÞ� þ ðβ þ δβÞ∇× ½∇ × ðus þ δusÞ� ¼ ds; (4)

where δus is the perturbed wavefield.By ignoring the high-order terms δα∇ð∇ · δusÞ and δβ∇ ×

ð∇ × δusÞ, and subtracting equation 3 from equation 4, we obtaina relation for the perturbed wavefield δus with respect to the modelperturbations in α and β:

δus − α∇ð∇ · δusÞ þ β∇ × ð∇ × δusÞ

¼ ½∇ð∇ · usÞ − ∇ × ð∇ × usÞ��δαδβ

�: (5)

The predicted data are extracted from the perturbed wavefield δus atthe receiver locations.We define an operator Q that, at each time and space posi-

tion, computes the dot product between vector ½∇ð∇ · usÞ−∇ × ð∇ × usÞ�T and the image m. Here, the first and second ele-ments of this vector used by operator Q are the decomposed P-and S-modes of the source wavefield us, respectively. The demigra-tion operator in equation 1 thus becomes

F ¼ KPQ; (6)

where P represents an elastic forward modeling operator that com-putes the perturbed wavefield δus for a source term Qm andK is anoperator that restricts the perturbed wavefield δus to the knownreceiver positions. Equation 6 maps the image m to the data dr,and its adjoint operator,

F⊤ ¼ Q⊤P⊤K⊤; (7)

maps the data dr to the image m; the operator K⊤ injects the re-corded data dr into the wavefield, and the adjoint wave-propagationoperator P⊤ computes the receiver displacement wavefieldur ¼ P⊤K⊤dr. Equation 7 describes a perturbation imaging condi-tion for elastic RTM because the application of the adjoint operatorF⊤ to the recorded data dr yields the images

δα ¼Xe;t

∇ð∇ · usÞ · ur; (8)

δβ ¼Xe;t

− ∇ × ð∇ × usÞ · ur; (9)

S316 Duan et al.

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

for the α and β models, respectively. From equations 8 and 9, we seethat images δα and δβ are computed by taking the zero-lag cross-correlation of elements used by operator Q with the displacementreceiver wavefield ur. These images indicate the model perturba-tions, instead of the reflection coefficients; therefore, they do notsuffer from polarity reversal, which is a common issue for elasticimages, whose values are related to angle-dependent reflectivity.Because our inversion algorithm updates δα and δβ images simul-taneously, we apply a scaling factor ϵ to the image δβ to balance theupdates of the two images. For the following synthetic examples,we use ϵ ¼ 1; for the field-data example, we estimate ϵ by compar-ing the root-mean-square values of gradients in the first iteration ofindependent inversions for δα and δβ. We use a conjugate directionsolver for inversion (Claerbout, 2014).

EXAMPLES

We show the results of our elastic LSRTM method using threeexamples. First, we run an inversion with a simple synthetic layeredmodel, and we analyze the relationships of model perturbations andelastic reflections. Second, we test the method using a more com-plex model. We use the elastic Marmousi model to generate pertur-bation models, background models, and a 2C data set. Finally, weapply our method to a real 2D ocean bottom cable (OBC) data setand compare the elastic RTM images with the LSRTM images.

Layered model

We use a simple example to demonstrate the algorithms for elas-tic migration. Each of the α and β models contains one horizontalreflector, but the two reflectors are at different depths, as shown inFigure 1a and 1b. We generate 30 2C shot gathers using a verticaldisplacement source with a 30 Hz peak frequency Ricker wavelet.Figure 2a–2f shows the x- and z-component snapshots of a wave-field with the source at (0.76, 0.06) km. The P-wave generates re-flections at the reflector in the αmodel, but not at the reflector in theβ model. Similarly, the S-wave generates reflections at the reflectorin the β model, but not at the reflector in the α model. The x- and z-components of this shot gather after direct wave removal are shownin Figure 3a and 3b, respectively. Note the four strong arrivals,which are, from top to bottom, PP, PS, SP, and SS reflections. Usingthe perturbation imaging condition (equation 6), we obtain the per-turbation images for δα and δβ shown in Figure 4a and 4b, respec-tively. Notice that additional reflectors appear in δα and δβ images,and these reflectors are generated by fake modes in the constructedreceiver wavefield.Figure 4c and 4d shows the LSRTM images after 10 iterations.

Compared with the RTM images (Figure 4a and 4b), the artifacts inthe LSRTM images are attenuated. Moreover, the peak values of theLSRTM images at x ¼ 0.6 km are closer to the amplitudes of thetrue perturbation compared with the values of the RTM images.Therefore, LSRTM improves elastic imaging with true amplitudeinformation and fewer artifacts, such as artifacts caused by the non-physical modes.

Marmousi model

The Marmousi II model (Martin et al., 2006) is fully elastic,which supports not only P-waves but also S-waves and con-verted waves. The model simulates hydrocarbon reservoirs that

dramatically decrease the value of α but slightly increase the valueof β. We smooth the original Marmousi II model horizontally andvertically to obtain the background models for α and β, shown inFigure 5a and 5b, respectively. We add a homogeneous layer at thetop of each model. Figure 6a and 6b shows the corresponding trueperturbation models for δα and δβ, respectively, which are incon-sistent in reservoir areas; e.g., only the δα model shows a reflectorwith a negative value at (2, 0.4) km. This inconsistency poses achallenge for elastic LSRTM, e.g., if the inversion allows a leakagebetween model parameters.We model 40 shots evenly spaced on the surface using a displace-

ment source with a 30 Hz peak frequency Ricker wavelet. The hori-zontal and vertical components of the source function have the sameamplitude to generate strong S-waves. The receiver spread is fixedfor all shots and spans from 0 to 3.0 km with 5 m sampling.The recorded data are modeled according to equation 7. The z-

component of one shot gather with the source location at (1.54,0.013) km is shown in Figure 7a. The arrival with high amplitudeis the reflection from the bottom of the homogeneous layer, and itsamplitude is much stronger than other arrivals in the recorded data.Thus, this arrival generates strong artifacts in the computed image,and the inversion mostly focuses on generating an image that best

a)

b)

Figure 1. Synthetic layered model example. (a) The αmodel with ahorizontal reflector at z ¼ 0.45 km, and (b) β model with a hori-zontal reflector at z ¼ 0.62 km.

Elastic LSRTM S317

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

matches these strong arrivals, instead of the late arrivals withweaker amplitudes. To obtain a more uniform update using all arriv-als, we use the data-weighting term W. The data-weighting matrixis the inverse of the envelope of the recorded data, which weightsdown the strongest arrivals. We also apply a smoothing operator inthe data space to the weighting function to avoid discontinuity alongthe time and space axes. Figure 7b shows the weighting functionin the data domain, which we compute from the shot gather(Figure 7a). Figure 8 shows the z-component of the weighted shotgather. Compared with the original recorded data (Figure 7a), the

amplitudes of the arrivals in the weighted data (Figure 8) are morebalanced.Figure 9a and 9b shows the RTM images for δα and δβ, respec-

tively. We observe that the events for the shallow reflectors in bothmodels have stronger amplitudes compared with those of thedeeper reflectors, and strong backscattering is present due tothe sharp interfaces in the background model. We apply an illu-mination compensation based on the source wavefield to theRTM images and LSRTM gradients, to balance nonuniform datacoverage.

a) b)

c) d)

e) f)

Figure 2. Synthetic layered model example. Snapshots of an elastic wavefield, for a source at coordinates (0.76, 0.06) km: (a) x- and (b) z-components of the wavefield at t ¼ 0.2 s; (c) x- and (d) z-components of the wavefield at t ¼ 0.3 s; and (e) x- and (f) z-components of thewavefield at t ¼ 0.4 s. Two horizontal lines indicate the locations of reflectors in α and β models. The P-waves generate reflections only at thetop reflector; and the S-waves generate reflections only at the bottom reflector.

S318 Duan et al.

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

a)

b)

c)

d)

Figure 3. Synthetic layered model example. For the source at(0.76, 0.06) km, (a) x- and (b) z-components of the recorded data;and (c) x- and (d) z-components of the data difference using theinverted model. The direct arrivals are removed in the shot gathers.Note that the predicted data match the recorded data in phase andamplitude.

a)

b)

c)

d)

Figure 4. Synthetic layered model example. (a) The δα and (b) δβimages after first iteration. In addition to the event at the correctdepth, there are two strong horizontal events in the δβ image, whichare the crosstalk artifacts. (c) The δα and (d) δβ images after 10iterations. Compared with the δα and δβ images after the first iter-ation, the artifacts have been attenuated.

Elastic LSRTM S319

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

The LSRTM images after 60 iterations are shown in Figure 10aand 10b. Figure 11a shows the convergence curve. The updated im-age for δβ has higher resolution than δα because, in general,S-waves have shorter wavelengths than P-waves, and we do notconsider attenuation in this experiment. The updated images areconsistent with the true perturbation images (Figure 6a and 6b).For example, only the δα image (Figure 10a) contains the reflectorwith negative value at (2, 0.4) km, which corresponds to a hydro-carbon reservoir in the true model that only decreases the value of α.The same weighting functions are applied to the two gathers. Fig-

ure 11b shows the z-component of data difference. The amplitudesof all arrivals in the data residual after inversion are small; i.e., formost arrivals, the predicted data match the recorded data in phaseand amplitude. Because we use Born data generated using the samewave equation as the recorded data, the amplitudes of the LSRTMimages match well the true perturbation models. In practice, witherrors in the background models and approximations of the waveequation, we may not obtain true-amplitude LSRTM images; how-ever, the relative amplitudes of the reflectors can still be estimatedcorrectly, which is beneficial for reservoir characterization.

Volve OBC data

We apply our elastic LSRTMmethod to data from the Volve field,located in the Norwegian North Sea. The Volve field has an averagewater depth of 90 m, and the field is characterized by a complexsubchalk reservoir (Szydlik et al., 2007). We are provided with

processed PP and PS data, recorded in an acquisition geometry con-sisting of 12 parallel receiver lines each with 240 receivers. How-ever, because PP and PS data typically are intended for independentPP and PS migrations, they are not well-suited for our inversionalgorithm, which requires full components of displacement data.For our test, we use a 2D line extracted from only the PP dataset, which we treat as the vertical component of the displacementvector. We band-pass the data between 0 and 15 Hz to attenuatehigh-frequency noise and to reduce the computational cost, andwe apply a gain in time to compensate for 3D effects. Figure 12ashows one shot gather at x ¼ 6 km, with the band-pass filter and 3Dcompensation applied. The arrivals at 2.5–3.0 s are PP reflectionsfrom the chalk layer.Additional processing is required to condition the data for elastic

LSRTM. Because S-waves were purposely attenuated in the pro-vided PP data in preparation for acoustic imaging, we use adata-weighting function (Figure 12b) for elastic imaging to attenu-ate predicted S-wave arrivals that otherwise would result in a largedata misfit. This data-weighting function also incorporates a maskto mute direct arrivals, as well as an additional weight that boostslater arrivals to balance reflector amplitudes in the migrationimages.The background α and β models are shown in Figure 13a and

13b, respectively, and they are computed from the provided P-and S-velocity models. We apply additional vertical and horizontalsmoothing to the original models, so that the background models

a)

b)

Figure 5. Synthetic Marmousi model example. Background (a) αand (b) β models. The receivers are at depth z ¼ 0.025 km, andthe sources are at depth z ¼ 0.013 km. The top layer is homo-geneous for both background models.

a)

b)

Figure 6. Synthetic Marmousi model example. True (a) δα and(b) δβ perturbation models. The perturbation models are not iden-tical; e.g., a reflector with negative value at (2, 0.4) km is presentonly in the δα model.

S320 Duan et al.

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

follow our assumption of homogeneous or slowly varying elasticmedia. A source wavelet for migration is estimated from thezero-offset data by matching the observed and predicted data usingthe background models. Using these models and an estimatedsource wavelet, we compute the RTM images shown in Figure 14aand 14b, in which we observe reflectors, as well as the migrationartifacts.For elastic LSRTM, we invert for δα and δβ images simultane-

ously. The scaling factor ϵ is 0.5, which is applied to δβ image to

balance the updates of the two images. Figure 15a and 15b showsthe LSRTM images after 12 iterations, at which point the conver-gence curve (Figure 16) becomes flat. Compared with the RTM im-ages, the LSRTM images have higher resolution and better balancedreflector amplitudes. In addition, shallow reflectors in the LSRTMimage for δβ (Figure 15b) are much more continuous and havelarger lateral extent along the x-direction, for example, atz ¼ 1.0 and 1.5 km. After the inversion converges, some reflectorsof less than z ¼ 3.5 km in the α image are eliminated; however,they appear as strong and continuous reflectors in the δβ image.Figure 17a and 17b shows the observed and predicted shot gathersat x ¼ 6 km, respectively, and Figure 17c shows the data differenceplotted on the same scale. The predicted data accurately reproducethe dominant reflection energy seen in the observed data, and theremaining residual in the data difference contains mostly noise andsome weaker reflection events. Some of these reflections with steepslopes are predicted S-waves that are missing in the observed data.Other weak reflections in the data residual may be due to anisotropyof the field, which is not considered in our current implementationof the algorithm.To evaluate the reflectors in the δα and δβ images, we generate

the predicted shot gathers using δα and δβ LSRTM images individu-ally. By comparing these shot gathers with the observed data(Figure 17a), we notice that the inverted δα image predictsmost of the arrivals in the observation with correct amplitudes(Figure 18a). The reflections predicted by the δβ image (Figure 18b)are relatively weaker than those predicted by the δα image.

a)

b)

Figure 7. Synthetic Marmousi model example. For the source lo-cation at (1.54, 0.013) km, (a) z-component of one shot gather. Thehighlighted arrival is the reflection from the bottom of the homo-geneous layer, and its amplitude is much stronger than other arrivalsin the recorded data. (b) The data weighting function generatedfrom the recorded data. The same weighting function is appliedto the horizontal and vertical components of the shot gather.

Figure 8. Synthetic Marmousi model example. For the source lo-cation at (1.54, 0.013) km, z-components of the weighted shotgather. Note that the amplitudes of all arrivals are more balancedcompared with Figure 7a.

a)

b)

Figure 9. Synthetic Marmousi model example. (a) The δα and(b) δβ RTM images with illumination compensation based onthe source wavefield.

Elastic LSRTM S321

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

a)

b)

Figure 10. Synthetic Marmousi model example. Updated (a) δαand (b) δβ images after 60 iterations. Note that the reservoir nearcoordinates (2.0, 0.4) km is correctly recovered in the updated δαimage, without any leakage in the updated δβ image.

a)

b)

Figure 11. Synthetic Marmousi model example. (a) The conver-gence curve. (b) The z-component of the data difference usingthe updated models (Figure 10a and 10b).

a)

b)

Figure 12. Volve data example. (a) One PP shot gather atx ¼ 6 km, which is assumed to be a vertical displacement gather.(b) Weighting matrix for this shot gather. The weighting functionmutes the direct arrivals, attenuates the predicted S-waves, andboosts later reflections.

a)

b)

Figure 13. Volve data example. Background (a) α and (b) βmodels.The top water layer is from z ¼ 0 to 90 m.

S322 Duan et al.

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Figure 18b shows mainly two types of arrivals: PP reflections withgentle slopes and converted-wave reflections with steep slopes. Theamplitudes of the predicted PP reflections are relatively strong at faroffset. This is to be expected because, according to Zoeppritz equa-tion, the β model in general affects the amplitudes of PP reflections,especially at far offsets. For example, the events at z ¼ 2.5–3.5 km

in the δβ image compensate for the amplitudes of PP reflections atz ¼ 2.5–3.0 s to match the observed data at far offsets.

Figure 19 shows enlarged views of data from within the dashedbox in Figure 17a: Figure 19a shows the observed data, Figure 19cshows data predicted using the δα and δβ images, Figure 19b showsdata predicted using only the δα image, and Figure 19d shows data

a)

b)

Figure 14. Volve data example. (a) The δα and (b) δβ RTM images.The chalk layer is from z ¼ 2.5 to 3.0 km. Note that the eventsabove the chalk layer in the δβ are weak and discontinuous.

a)

b)

Figure 15. Volve data example. (a) The δα and (b) δβ LSRTM im-ages after 12 iterations. Compared with the RTM images (Figure 14aand 14b), the LSRTM images have higher resolution and better bal-anced amplitudes. The reflectors at greater than 1.5 km appear in theδβ image after inversion with larger lateral extent along the x-direc-tion than those in the δα image.

Figure 16. Volve data example. The convergence curve.

Figure 17. Volve data example. The weighted (a) observed data,(b) predicted data, and (c) data residual after 12 iterations. The pre-dicted data accurately reproduce the dominant reflection energyseen in the observed data. The data residual contains mostly noiseand some weaker reflection events.

Elastic LSRTM S323

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

predicted using only the δβ image. The dashed line in each panelmarks the same arrival, and this arrival is one that was attenuatedduring processed of the PP data. Notice that because the δα imagepredicts this arrival (shown in the bottom left panel), the δβ imagetries to predict the same arrival but with opposite polarity (shown inthe bottom right panel), so that the total prediction agrees with theobservation. Had this (and other) arrivals not been removed fromthe observed data, we would expect the data predicted from the

δα and δβ images, as well as the corresponding reflector amplitudes,to change accordingly. This is an illustration of how processing cannegatively affect LSRTM.We also test the inversion algorithm using PP data and PS data,

which were treated as the vertical and horizontal components of thedisplacement vector, respectively. However, we were not able topredict both components with correct phases and amplitudes. Weattribute this outcome to the fact that the data were processedfor independent PP and PS imaging, and thus the arrivals in thePP and PS gathers might not be consistent with each other, in termsof phase and amplitude; therefore, this data set does not fit our elas-tic LSRTM algorithm underlying assumptions, which require ver-tical and horizontal displacement data, instead of separated PP andPS gathers.

CONCLUSION

We propose a method for elastic LSRTM using a perturbationimaging condition. The images computed using our method re-present perturbations of squared P and S velocities, and they donot suffer from polarity reversals, which usually characterize con-ventional elastic images for converted waves. The synthetic andfield data tests of the perturbation imaging condition demonstratethat elastic LSRTM produces images with fewer migration artifactsand higher resolution compared with the corresponding RTM im-age. The field data example also shows the importance of properpreprocessing of the recorded data, which should provide theLSRTM algorithm with data consistent with the output of the Bornmodeling operator.

ACKNOWLEDGMENTS

We thank the sponsors of the Center for Wave Phenomena, whosesupport made this research possible. We also thank Statoil ASA andthe Volve license partners ExxonMobil E&P Norway AS andBayerngas Norge AS, for the release of the Volve data. The viewsexpressed in this paper are the views of the authors and do not nec-essarily reflect the views of Statoil ASA and the Volve field licensepartners. The reproducible numeric examples in this paper use theMadagascar open-source software package (Fomel et al., 2013)freely available from http://www.ahay.org.

REFERENCES

Artman, B., I. Podladtchikov, and A. Goertz, 2009, Elastic time-reversemodeling imaging conditions: 79th Annual International Meeting,SEG, Expanded Abstracts, 1207–1211.

Claerbout, J. F., 1992, Earth soundings analysis: Processing versus inver-sion: Blackwell Scientific Publications.

Claerbout, J., 2014, Geophysical image estimation by example, www. lulu.com, ISBN 978-1-312-47467-3.

Dai, W., and G. T. Schuster, 2013, Plane-wave least-squares reverse-time mi-gration: Geophysics, 78, no. 4, S165–S177, doi: 10.1190/geo2012-0377.1.

Dai, W., X. Wang, and G. T. Schuster, 2011, Least-squares migrationof multisource data with a deblurring filter: Geophysics, 76, no. 5,R135–R146, doi: 10.1190/geo2010-0159.1.

Denli, H., and L. Huang, 2008, Elastic-wave reverse-time migration with awavefield-separation imaging condition: 78th Annual International Meet-ing, SEG, Expanded Abstracts, 2346–2350.

Dickens, T. A., and G. A. Winbow, 2011, RTM angle gathers using Poyntingvectors: 81st Annual International Meeting, SEG, Expanded Abstracts,3109–3113.

Dong, S., J. Cai, M. Guo, S. Suh, Z. Zhang, B. Wang, and Z. Li, 2012, Least-squares reverse time migration: Towards true amplitude imaging and im-

a)

b)

Figure 18. Volve data example. Predicted shot gathers using only(a) δα and (b) δβ LSRTM images. The δα image predicts most of thearrivals in the observation with correct amplitudes. The δβ imagecompensates for the amplitude variation with offset effect of thePP reflections at far offsets (e.g., x ¼ 8.5 km, t ¼ 3.0 s).

a) b)

c) d)

Figure 19. Volve data example. (a) Magnified views of theweighted observation, (c) prediction computed using δα and δβ im-ages, (b) prediction computed using only δα image, and (d) predic-tion computed using only δβ image. A different clip from theprevious plots (Figures 17a and 18b) is applied to the magnifiedpanels for display. The box in Figure 17a indicates the magnifiedarea. The dashed line highlights the same arrival in each panel. Theδα and δβ images predict the same arrival with opposite polarities,so that the total prediction agrees with the observation.

S324 Duan et al.

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

proving the resolution: 82nd Annual International Meeting, SEG, Ex-panded Abstracts, doi: 10.1190/segam2012-1488.1.

Du, Q., X. Gong, Y. Zhu, G. Fang, and Q. Zhang, 2012a, PS wave imagingin 3D elastic reverse-time migration: 82nd Annual International Meeting,SEG, Expanded Abstracts, doi: 10.1190/segam2012-0107.1.

Du, Q., Y. Zhu, and J. Ba, 2012b, Polarity reversal correction for elasticreverse time migration: Geophysics, 77, no. 2, S31–S41, doi: 10.1190/geo2011-0348.1.

Duan, Y., and P. Sava, 2015, Scalar imaging condition for elastic reversetime migration: Geophysics, 80, no. 4, S127–S136, doi: 10.1190/geo2014-0453.1.

Duan, Y., and P. Sava, 2014, Elastic reverse-time migration with OBS multi-ples: 84th Annual International Meeting, SEG, Expanded Abstracts,4071–4076, doi: 10.1190/segam2014-1395.1.

Fomel, S., P. Sava, I. Vlad, Y. Liu, and V. Bashkardin, 2013, Madagascar:Open-source software project for multidimensional data analysis andreproducible computational experiments: Journal of Open Research Soft-ware, 1, e8, doi: 10.5334/jors.ag.

Hokstad, K., R. Mittet, and M. Landrø, 1998, Elastic reverse time migrationof marine walkaway vertical seismic profiling data: Geophysics, 63,1685–1695, doi: 10.1190/1.1444464.

Hou, J., and W. W. Symes, 2015, An approximate inverse to the extendedBorn modeling operator: Geophysics, 80, no. 6, R331–R349, doi: 10.1190/geo2014-0592.1.

Huang, Y., and G. T. Schuster, 2012, Multisource least-squares migration ofmarine streamer and land data with frequency-division encoding: Geophysi-cal Prospecting, 60, 663–680, doi: 10.1111/j.1365-2478.2012.01086.x.

Hudson, J., and J. Heritage, 1981, The use of the Born approximation inseismic scattering problems: Geophysical Journal International, 66,221–240, doi: 10.1111/j.1365-246X.1981.tb05954.x.

Jaramillo, H. H., and N. Bleistein, 1999, The link of Kirchhoff migration anddemigration to Kirchhoff and Born modeling: Geophysics, 64, 1793–1805, doi: 10.1190/1.1444685.

Kaplan, S. T., P. S. Routh, and M. D. Sacchi, 2010, Derivation of forwardand adjoint operators for least-squares shot-profile split-step migration:Geophysics, 75, no. 6, S225–S235, doi: 10.1190/1.3506146.

Kuehl, H., and M. Sacchi, 2002, Robust AVP estimation using least-squareswave-equation migration: 72nd Annual International Meeting, SEG, Ex-panded Abstracts, doi: 10.1190/1.1817231.

Martin, G. S., R. Wiley, and K. J. Marfurt, 2006, Marmousi2: An elasticupgrade for Marmousi: The Leading Edge, 25, 156–166, doi: 10.1190/1.2172306.

Nemeth, T., C. Wu, and G. T. Schuster, 1999, Least-squares migration of in-complete reflection data: Geophysics, 64, 208–221, doi: 10.1190/1.1444517.

Patrikeeva, N., and P. Sava, 2013, Comparison of angle decompositionmethods for wave-equation migration: 83rd Annual International meeting,SEG, Expanded Abstracts, 3773–3778.

Ribodetti, A., S. Operto, W. Agudelo, J.-Y. Collot, and J. Virieux, 2011,Joint ray + Born least-squares migration and simulated annealing optimi-zation for target-oriented quantitative seismic imaging: Geophysics, 76,no. 2, R23–R42, doi: 10.1190/1.3554330.

Rocha, D., N. Tanushev, and P. Sava, 2016, Isotropic elastic wavefield im-aging using the energy norm: Geophysics, 81, no. 4, S207–S219, doi: 10.1190/geo2015-0487.1.

Schuster, G. T., 1993, Least-squares cross-well migration: 63rd AnnualInternational Meeting, SEG, Expanded Abstracts, 110–113.

Stanton, A., and M. Sacchi, 2015, Least squares wave equation migration ofelastic data: 77th Annual International Conference and Exhibition, EAGE,Extended Abstracts, N116.

Sun, R., and G. A. McMechan, 1986, Pre-stack reverse-time migration forelastic waves with application to synthetic offset vertical seismic profiles:Proceedings of the IEEE, 74, 457–465, doi: 10.1109/PROC.1986.13486.

Sun, R., G. A. McMechan, C. Lee, J. Chow, and C. Chen, 2006, Prestackscalar reverse-time depth migration of 3D elastic seismic data: Geophys-ics, 71, no. 5, S199–S207, doi: 10.1190/1.2227519.

Szydlik, T., P. Smith, S. Way, L. Aamodt, and C. Friedrich, 2007, 3D PP/PSprestack depth migration on the Volve field: First Break, 25, 43–47.

Trad, D., 2015, Least squares Kirchhoff depth migration: Implementation,challenges, and opportunities: 85th Annual International Meeting, SEG,Expanded Abstracts, 4238–4242, doi: 10.1190/segam2015-5833655.1.

Wong, M., B. L. Biondi, and S. Ronen, 2015, Imaging with primaries andfree-surface multiples by joint least-squares reverse time migration: Geo-physics, 80, no. 6, S223–S235, doi: 10.1190/geo2015-0093.1.

Wu, R., R. Yan, X. Xie, and D. Walraven, 2010, Elastic converted-wave pathmigration for subsalt imaging: 80th Annual International Meeting, SEG,Expanded Abstracts, 3176–3180.

Yan, J., and P. Sava, 2008, Isotropic angle-domain elastic reverse-time migra-tion: Geophysics, 73, no. 6, S229–S239, doi: 10.1190/1.2981241.

Elastic LSRTM S325

Dow

nloa

ded

06/2

1/17

to 1

38.6

7.12

9.34

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/